Can a small child of 3 get shingles?

Yes: I have seen it as early as 15 months. All it takes is a past infection with chickenpox or the chickenpox vaccine. Statistically the vaccine has less shingles than the wild virus but it can happen. Reactivation in early childhood follows the same pattern that is does in adults although the subjective complaints of pain prior to eruption seems to be less prominent.

Shingles: Children who had chickenpox before age 1, whose mother had it late in pregnancy or who are immunocompromised have a higher risk for developing shingles. REF: https://www.stlouischildrens.org/health-resources/pulse/understanding-shingles